Application of Newton’s polynomial interpolation scheme for variable order fractional derivative with power-law kernel

This paper, offers a new method for simulating variable-order fractional differential operators with numerous types of fractional derivatives, such as the Caputo derivative, the Caputo–Fabrizio derivative, the Atangana–Baleanu fractal and fractional derivative, and the Atangana–Baleanu Caputo derivative via power-law kernels. Modeling chaotical systems and nonlinear fractional differential equations can be accomplished with the utilization of variable-order differential operators. The computational structures are based on the fractional calculus and Newton’s polynomial interpolation. These methods are applied to different variable-order fractional derivatives for Wang–Sun, Rucklidge, and Rikitake systems. We illustrate this novel approach’s significance and effectiveness through numerical examples.

The paper is organized as follows: the basic definitions of variable-order fractional derivatives are studied in Section "Preliminaries".In Section "New numerical approach", see about the new numerical scheme of variable order derivative.Various fractional derivatives of variable order are investigated in Section "Variable order fractional derivative", and some validation of the numerical schemes is presented.In Section "Stability analysis", study about the stability analysis of Rikitake dynamical system.In Section "Comparative study", there is a there is a comparison study about the four distinct variable-order fractional derivatives.Finally, the conclusion is described in the last section.

Preliminaries
In this section the basic definition of variable order distinct fractional derivatives are discussed.Definition 1.1 5 The Caputo fractional derivative with variable order α(τ ) is defined as where Ŵ(α(τ )) is Gamma function.
+ g(τ p ,x p )−2g(τp−1,x p−1 )+g(τp−2,x p−2 ) . Then we can rearrange the above equation and we get the two step Newton's approximation approach w h e r e

Atangana-Baleanu-Fractal fractional derivative
Consider the general Cauchy problem of Atangana-Baleanu fractal fractional derivative By applying the definition of fractal fractional integral, Then, (3)

Variable order fractional derivative
In this section, the various type of variable order fractional derivatives for distinct chaotic attractor with new numerical schemes are presented. x x p+1 =x(0 Table 1.Parameter values and initial conditions for the chaotic systems.

Systems Initial conditions Parameter values
Wang-Sun Rucklidge u 0 = 0.01, v 0 = 0.02, w 0 = 0.01 y 1 = 2, y 2 = 6.7 Rikitake u 0 = 0.8, v 0 = 0.9, w 0 = 0.5 Vol.:(0123456789)In recent times, several novel chaotic systems have emerged, notably the Chen system, the generalized Lorenz system family, and the hyperbolic-type generalized Lorenz canonical form 46,47 .In reality, the Rossler system with a single wing exhibits topological and dynamic simplicity compared to the generalized Lorenz system, which features a double wing.It is evidently important to explore the creation of more intricate chaotic systems incorporating multi-scroll or multi-wing attractors, both in theoretical studies and practical experiments with engineering applications.
From both theoretical and practical perspectives, the preference is consistently for a lower-dimensional system characterized by a simpler algebraic structure yet one that exhibits more intricate topological features, including multi-wings, broader frequency bandwidths, and complex dynamics involving rich bifurcations.The question naturally arises as to whether a straightforward 3-D quadratic autonomous system can generate a genuine fourwing chaotic attractor with a wide frequency spectrum.This inquiry was addressed by Liu and Chen (2003), who initially suggested it was not feasible.However, in subsequent research, we demonstrate the existence of such a system, affirming that it can be implemented using physical circuits.Therefore, consider the 3-D four-wing smooth autonomous chaotic attractor of the Wang-Sun chaotic system: The employed method for obtaining (6) enables the reconstruction of asymptotically precise solutions for the partial differential equations (PDEs), derived from solutions of ( 6) with an error approaching zero.Specifically, the presence of chaotic trajectories in the Rucklidge system indicates the presence of chaotic trajectories in the PDEs.Therefore, our system serves as a legitimate representation of the PDEs within a parameter regime that has not undergone thorough examination.Nevertheless, this particular regime holds physical significance, as convection in a robust vertical magnetic field is known to occur in narrow rolls.
The Rucklidge chaotic system, distinct from the generalized Lorenz system, is a three-dimensional chaotic system resembling Lorenz.It serves as a representation of a dual-convection process.Therefore, the Rucklidge model characterizes the convection phenomenon in a horizontal layer of Boussinesq fluid, incorporating lateral constants to elucidate the convection dynamics, particularly at the juncture where chaotic solutions manifest.So consider the Rucklidge chaotic system: The Rikitake system, derived through experimentation with a two-disk dynamo apparatus, represents a threedimensional vector field.This system serves as a model for the geomagnetic field, providing insights into the observed irregular switch in its polarity.Characterized by a 3-dimensional Lorenz-type chaotic attractor centered around its two singular points, the Rikitake system lacks confinement within an ellipsoidal surface, distinguishing it from the Lorenz attractor.Geophysicists acknowledge that the Earth's magnetic field has undergone numerous polarity reversals throughout its geological history.One common mechanical model employed to investigate these reversals is a two-disk dynamo system introduced by Rikitake 48 .In our Rikitake system (7), the notation z 1 denotes resistive dissipation, while the parameter z 2 signifies the variance in angular velocities between two dynamo discs.
The system comprises a pair of interconnected Faraday-disk dynamos, both identical and of the Bullard type.The dynamics of this system are governed by a set of three-dimensional nonlinear differential equations: The initial conditions and system parameter values for the Wang-Sun, Rucklidge, and Rikitake chaotic systems are shown in Table 1 for simple understanding.

Remark 3.1
The integer-order dynamical system (7) is obviously unstable.However, the fractional order and variable order dynamical system is stable for any fractional and variable order 0 < α(τ ) < 1.
Figure 1 presents the fractional order Caputo derivative.This figure represents the system's periodic behavior for the fractional order α(tau) ∈ (0, 1) .Figure 1a is an integer-order unstable dynamical system, and Fig. 1b-d represents the Caputo derivative with various fractional orders.

Numerical approach for Caputo fractional variable order derivative
In this subsection discuss the new type of numerical schemes for the variable order Caputo sense fractional derivative of Lorenz family attractions like Wang-Sun system, Rucklidge system, and Rikitake system.
The general solution can be applied in our chaotic system, as follows (5)

Variable order Wang-Sun chaotic system
Consider the Wang-Sun chaotic system of Caputo fractional variable order system In Fig. 2, represent the chaotic behavior of Wang-Sun attraction with variable order α(τ ) = tanh(τ + 1) .In Fig. 2a-d presents the chaotic nature of 3D state phase plane of uvw, uv-phase plane, vw-phase plane, uw-phase plane respectively.

Variable order Rucklidge chaotic system
Consider the Rucklidge chaotic system of Caputo fractional variable order system In Fig. 3, we represent the chaotic behavior of the variable-order Caputo fractional derivative of Rucklidge attraction with order α(τ ) = tanh(τ + 1) .Figure 3a-d present the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.
Variable order Rikitake chaotic system Consider the Rikitake chaotic system of Caputo fractional variable order system In Fig. 4, study the chaotic behavior of the Rikitake attractor with order α(τ ) = tanh(τ + 1) with variable order Caputo derivative sense.Figures 4a-d present the chaotic behavior of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.

Numerical approach for Caputo-Fabrizio fractional variable order derivative
The general solution can be applied in our chaotic system, as follows ( 9) www.nature.com/scientificreports/Variable order Wang-Sun chaotic system Consider the Wang-Sun chaotic system of Caputo-Fabrizio fractional variable order system Figure 5 presents the chaotic behavior of Wang-Sun attraction with order α(τ ) = tanh(τ + 1) .Figures 5a-d pre- sent the variable-order Caputo-Fabrizio fractional derivative with the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.In this derivative, dynamical behavior is very slow when applied at t = 200 .Moreover, to consider at time t = 400 , the dynamical nature is the same as the other fractional derivatives like Caputo, Caputo-Frabrizio, and Atangana-Baleanu-Caputo derivatives.

Variable order Rucklidge chaotic system
Consider the Rucklidge chaotic system of Caputo-Fabrizio fractional variable order system Figure 6 presents the chaotic behavior of Rucklidge attraction with order α(τ ) = tanh(τ + 1) .Figures 6a-d present the variable-order Caputo-Fabrizio fractional derivative with the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, uw-phase plane respectively.
Variable order Rikitake chaotic system Consider the Rikitake chaotic system of Caputo-Fabrizio fractional variable order system Figure 7 presents the chaotic behavior of Rikitake attraction with order α(τ ) = tanh(τ + 1) .Figures 7a-d present the variable-order Caputo-Fabrizio fractional derivative with the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively. , ,

Numerical approach for Atangana-Baleanu-Caputo fractional variable order derivative
In this section, discuss the Atangana-Baleanu-Caputo fractional variable order derivative of some distinct dynamical systems.
The general solution can be applied in our chaotic system, as follows, Variable order Wang-Sun chaotic system Consider the Wang-Sun chaotic system of Atangana-Baleanu-Caputo fractional variable order system Figure 8 presents the chaotic behavior of Wang-Sun attraction with order α(τ ) = tanh(τ + 1) .Figures 8a-d present the variable-order Atangana-Baleanu-Caputo fractional derivative with the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.

Variable order Rucklidge chaotic system
Consider the Rucklidge chaotic system of Atangana-Baleanu-Caputo fractional variable order system Figure 9 presents the chaotic behavior of Rucklidge attraction with order α(τ ) = tanh(τ + 1) .Figures 9a-d present the variable-order Atangana-Baleanu-Caputo fractional derivative with the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.
Variable order Rikitake chaotic system Consider the Rikitake chaotic system of Atangana-Baleanu-Caputo fractional variable order system ( 14)  .Figures 10a-d present the variable-order Atangana-Baleanu-Caputo fractional derivative with the chaotic nature of the 3D state phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.

Numerical approach for Atangana-Baleanu fractal fractional variable order derivative
In this subsection, the newly introduced integral and differential operator of various chaotic attractor are considered .That type of derivative is called the fractal and fractional derivative.Fractal fractional order derivatives offer a unique and powerful tool for modeling complex systems with non-local and memory-dependent behaviors, providing several advantages.Firstly, they offer a more nuanced representation of dynamical systems compared to traditional integer-order derivatives, capturing intricate features such as long-range correlations, self-similarity, and multi-fractal properties present in many natural phenomena.This enables more accurate and comprehensive modeling of complex systems across various fields, including physics, biology, finance, and engineering.Additionally, fractal fractional order derivatives exhibit superior adaptability to non-stationary and non-linear processes, making them particularly suitable for describing phenomena with evolving dynamics or irregular patterns.Moreover, their fractional nature facilitates the incorporation of memory effects, enabling the modeling of systems with memory-dependent behaviors, which are prevalent in many real-world applications.
The general solution can be applied in our chaotic system, as follows, ( 16) Figure 11 represents the chaotic behavior of variable-order Wang-Sun attraction with order α = 1 and β(τ ) = tanh(τ + 1) .Figures 11a-d present the variable-order Atangana-Baleanu fractal-fractional derivative with the chaotic nature of the 3D phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.
Variable order Rucklidge chaotic system Consider the Rucklidge chaotic system of Atangana-Baleanu Fractal fractional variable order system Figure 12 represents the chaotic behavior of variable-order Rucklidge attraction with order α = 1 and β(τ ) = tanh(τ + 1) .Figures 12a-d present the variable-order Atangana-Baleanu fractal-fractional derivative with the chaotic nature of the 3D phase planes of uvw, uv-phase plane, vw-phase plane, and uw-phase plane, respectively.

Stability analysis
In this section, we discussed the Hopf bifurcation and Lyapunov exponents of the fractional order Rikitake dynamical system.The corresponding linearized system is defined as follows with equilibrium points (u * , v * , w * ).
In order to discuss the local stability results of system (20), we take the Laplace transform on both sides of (20).

Comparative study
In this section, a comparative study about various aspects of the numerous derivatives, like the fractional variable order Caputo derivative, the Caputo Fabrizio derivative, the Atangana-Baleanu derivative, and finally the Atangana-Baleanu fractal and fractional derivative of distinct chaotic systems, are discussed.

Wang-Sun dynamical system
In this subsection, consider the four useful derivatives in real-life scenarios for the Wang-Sun chaotic system.

Rikitake dynamical system
In this subsection, consider the four useful derivatives in real-life scenarios for the Rikitake chaotic system.Figures 21,22 and 23 show the various distinct variable-order derivatives that are applied and validated for the Rucklidge dynamical systems.In Fig. 21, the variable-order fractional derivative is presented.In Fig. 21a, the Caputo fractional variable order derivative, and Fig. 21b, the Caputo-Fabrizio fractional variable order derivative, Fig. 21c shows the Atangana-Baleanu-Caputo derivative, and Fig. 21d displays the Atangana-Baleanu fractal-fractional variable order derivatives with order α = 1 , β(τ ) =     .In Fig. 22, the variableorder fractional derivative is presented.Figures 22a-d represent the 3D chaotic behavior of the state equation (u, v, w) with variable order α(τ ) = 0.98 + 0.005 sin(τ π/8) .In Fig. 23, the variable-order fractional derivative is presented.Figure 23a-d

Conclusion
This study analyzed numerous variable orders for unique chaotic systems utilizing Newton's polynomial interpolation, which was proposed for presenting computational solutions for fractional chaotic systems with power kernels with each variable order.In this work, the researchers established a numerical approach to deal with several chaotic challenges.We are certain of the method's significant worth and the existence of several popular numerical schemes.The variable-order Adams-Bashforth approach has restrictions that are strengthened by the introduction of the alternative numerical method.The process is straightforward, effective, and precise.Moreover, a tiny discretization step is not required, compared to the Adams-Bashforth approach, which lowers computational effort.The new type of numerical approximation is applied and validated for the different chaotic systems with various variable-order fractional derivatives.In the future, this approximation will be applied to time-varying and time-independent delay systems in various real-life scenarios.Also, the chaotic behavior of the fractional dynamical systems using the Poincare map will be considered.

Figure 1 .
Figure 1.Integer and non-integer order Caputo derivative of Rikitake dynamical system.

Figure 5 .
Figure 5. Numerical schemes for the variable order Caputo-Fabrizio fractional derivative of Wang-Sun system.

Figure 13 .
Figure 13.Numerical simulation of variable order Atangana-Baleanu fractal and fractional derivative of Rikitake equations.
represent the 3D chaotic behavior of the state equation (u, v, w) with variable order α(τ ) = 1.
represent the 3D chaotic behavior of the state equation (u, v, w) with variable order α(τ ) = 1.